Normalized Solutions Of Two Classes Of Schr(?)dinger Type Equations With Nonlocal Terms

In this thesis.by using the variational method and the elliptic equation theory.we mainly study the existence.multiplicity and asymptotic behavior of normal solutions for two classes of Schrodinger type equations with non-local terms.Our work includes the following two parts:In Chapter 2.we consider a Schr(?)dinger type equation with the Hardy potential and a non-local term,whose normal solutions can be transformed to find critical points of the corresponding functional restricted on some constraint set.By choosing an appropriate constraint set.a special path and introducing an auxiliary functional.we define a sequence of minimax values of the energy functional corresponding to the Schr(?)dinger equation.and prove that these minimax values are the critical point values of the functional restricted on the constraint set.Also we prove that the sequence of the critical point values is unbounded via the idea of the fountain theorem,which indicates that the Schrodinger equation has infinitely many normal solutions.Moreover,we discuss the asymptotic behavior of normal solutions for the equation.This result characterizes the relationship between normal solutions of the Schrodinger type equations with and without nonlocal terms.This result is a supplement to the results obtained by Wang Zhengping et al.when studying three-dimensional situations where the nonlinear term is a special power term,and for the first time,we provide asymptotic convergence results for the normal solutions of this Schr(?)dinger type equations.In Chapter 3.we consider a nonlinear Choquard equation which is also a Schr(?)dinger type equation with a non-local term.By variational methods,we can transform the existence of the normalized solutions to the compactness of the minimizing sequences restricted on some constraint for the corresponding functional to the equation.By choosing an appropriate constraint set,using the techniques and methods such as the principle of concentration-compactness,the truncation function technique and the discrimination signs for the Lagrange multipliers,we systematically study the influence of the structure of nonlocal terms to the energy functional,and we verify the compactness of the minimization sequence of the constraint minimization problem,and therefore we obtain a nonexistence result of the normalized solutions and existence results of ground state normalized solutions for the Choquard equation.